# [SOLVED] Rearranging a differential equation

• Nov 25th 2008, 07:21 AM
Bud
[SOLVED] Rearranging a differential equation
Hi I am supposed to "just" rearrange the following equation and at one point substitute dh/dt = A/rho with the constant A:

http://www.texify.com/img/%5CLARGE%5...7D%7Bdh%7D.gif
With the constant C and rho(i).

The result looks like this:

http://www.texify.com/img/%5CLARGE%5...-%5Crho%29.gif

I cannot rearrange the 1st equation in a way that I obtain the 2nd. I assume that one has to ibntegrate at one point to get rid of the logarithm though.

If $\displaystyle \rho=f(h)$ and $\displaystyle h=g(t)$ then $\displaystyle \frac{d\rho}{dt}=\frac{d\rho}{dh}\frac{dh}{dt}$ right? So if we calculate what $\displaystyle \frac{d\rho}{dh}$ is and multiply by the given $\displaystyle \frac{dh}{dt}=\frac{A}{\rho}$, that would give us $\displaystyle \frac{d\rho}{dt}$. So what is:
$\displaystyle \frac{d}{dh}\left\{\ln\left(\frac{\rho}{\rho_i-\rho}\right)\right\}$
You can figure that out. Then isolate the $\displaystyle \frac{d\rho}{dh}$ term and then just multiply by $\displaystyle \frac{A}{\rho}$